Indestructibility, measurability, and degrees of supercompactness

Abstract

AbstractSuppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ+ supercompact} is unbounded in κ. If in addition λ is 2λ supercompact, then A2 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is δ+ supercompact} is unbounded in κ as well. The large cardinal hypotheses on λ are necessary, as we further demonstrate by constructing via forcing two distinct models in which either \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A_1 = \varnothing$\end{document} or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A_2 = \varnothing$\end{document}. In each of these models, there is an indestructibly supercompact cardinal κ, and a restricted large cardinal structure above κ. If we weaken the indestructibility requirement on κ to indestructibility under partial orderings which are both κ‐directed closed and (κ+, ∞)‐distributive, then it is possible to construct a model containing a supercompact cardinal κ witnessing this degree of indestructibility in which every measurable cardinal δ < κ is (at least) δ+ supercompact.